If a CRN has only one dynamical degree of freedom, a discrete
Laplacian acting on its moments reduces to a recursion relation on the
moment ratios. We showed
in [1]}, [2]} that such
recursions can be solved by a method of matched asymptotic expansions,
in which low-order modes are solved recursively as one would expect,
upward from the mean or variance, but high-order modes are solved
(counterintuitively) by anchoring the moment ratios at large \(k\) ,
where they converge asymptotically to values set by parameters in the
generator. For multivariate distributions, however, the corresponding
method of solution encounters the difficulty faced in the solution of
other Laplacian equations, which is the specification of boundary
conditions. For the CRN moment hierarchies, the difficult boundaries
are those on which one or more of the moments is not of
asymptotically large order, since there is no approximation scheme
which gives an estimate of such moments.
